CHAP. VIII. First Solution of the Problem of the Three Bodies under its most simple Conditions: that is, when the Body, previously to the Action of the Disturbing Force, is supposed to revolve in an Orbit without Eccentricity and Inclination; the Orbit, changed by the Action of the Disturbing Force, not strictly Elliptical. THE instances in the preceding Chapter were intended principally to explain the cause of that introduction of the arcs of a circle which renders faulty the expression of the radius vector. They have served that end, and the purpose of illustration, as well as more complex instances would have done. But they are altogether fictitious and hypothetical, since they exclude, besides other conditions, the essential one of a tangential disturbing force. The results of the ninth Section of the Principia of Newton have been compared and made to correspond with certain peculiar integral values of the differential equation (see pp. 109, 121.). In both cases, there is the same supposition with regard to the disturbing force. In the ninth Section, Newton's Extraneous Force, as it is there called, acts solely in the direction of the radius: and the disturbing force has been expounded by this equation, K Two cases have been considered, when n = 1, and when n=3, (see pp. 109, 121.) that is, when the disturbing force varies inversely as the cube of the distance, and when A varies as the distance. In the former, the exact value of u is expressed by an equation, such as whatever e be; or, [which is Newton's mode of considering the subject, (see Prop. XLIV.)] the body's place can always be found, and exactly, in a moveable ellipse, whatever its eccentricity be. In the latter case, when the disturbing force varies as the distance, an equation, such as approximately represents the value of u; and that, only when e is very small: or, according to Newton, (ninth Section, Prop. XLV.) the body's place may nearly be found in a moveable ellipse, when the orbit's eccentricity is very small; and the like equations and constructions obtain approximately for all other values of **. It has been already remarked, (p. 122.) that some mathematicians, persuaded that Newton meant to find the progression of the Lunar Apogee by the method of the ninth Section, have pursued that method. Now in that Section there is no tangential disturbing force, and, besides, the expression for that part of the Sun's disturbing force which acts in the direction of the radius, (see p. 60.) is unlike Newton's +CA). It was necessary for them, therefore, to shew by some probable arguments, that, in a problem of such importance as that of the Lunar Apogee, the former force could be dispensed with, and that the latter might be reduced to Newton's form. Now, with regard to the first point; the tangential force T (see p. 60.) is 3 m'r is very small. other: since, if 3 m'r 23 sin. 2w, which, from the largeness of the denominator, But, besides its smallness, its effects counteract each 3 m'r 23 3 m'r sin. 2 accelerate the body, sin. (180+2)= sin. 2 equally retards the body; which counteraction (since may be any angle) must accordingly take place for all corresponding points of the orbit. The mean effect therefore of this force, it was pre sumed, It has been just remarked that the instances of the preceding Chapter, framed for the purpose of illustration, are, with reference to the real circumstances in nature, fictitious and hypothetical. But, we may add to this remark, every instance which can be given is, to a certain degree, hypothetical. The inefficiency of the art of calculation obliges us to suppose a greater simplicity in the conditions of our problems than exists. The kind of simplification, however, which will be given to the succeeding instances is different from that which the preceding possess. Instead of excluding altogether the tangential force, its sumed, (not rightly inferred) would not materially affect the progression of the apsides. With regard to the second point, we have, by p. 60. If we substitute in the last term 180 2w instead of 2w, it becomes (see Trig. p. 28.) the value of P, therefore, is as much increased by the last term, in this situation of the body, as it was diminished in the former, and, since the same holds whatever be, that is, since the same result is true for every point in the orbit, the last term cos. 2 w is said, during an 3 m'r entire revolution, to be destroyed by the opposition of signs. Under this explanation, then, the mean force may be said to be and, if that force alone operates, the equation would be (see p. 111.), approximate value will be assumed and substituted in II: and, of the force that acts in the direction of the radius, all the essential terms at least, will be retained, although in determining their coefficients many small quantities will be rejected. And thus it shall happen, that the results will not be altogether remote from the truth, but will accord, in some degree, with the observed phenomena. If we look to the History of the Problem of the Three Bodies, it exhibits a series of solutions successively more and more exact. The Calculus, which was the instrument of solution, grew up with Physical Astronomy, and, as it advanced, additional conditions were introduced into the problem; so that, as the fruit of time, Laplace's Theory of the Moon, (without any reference to the genius of the two authors) is necessarily more perfect than Clairaut's. The present business of this Treatise, however, is not with the most complete solutions. Intended to serve as an introduction to Physical Astronomy, it will begin with the most simple cases, and be guided, very nearly, by their historical order. But, even according to this plan, there are two ways of proceeding. We may either select what are, in fact, the most simple cases in Nature, or we may, by hypothesis, simplify the conditions of some of the more complex cases. For instance, Venus revolving in a nearly circular orbit, and disturbed by a body as remote as Saturn, the plane of whose orbit is very little inclined to that of Venus's, is nearly as simple a case as is that of the Moon, when, as in the first essays of solution, that planet is supposed to revolve in an orbit circular, and coincident with the plane of the ecliptic. This regards the analytical difficulty of solution; but, with reference to arithmetical exactness, it is plain that the results in the first case, when specific numbers are substituted, must be more conformable to observation than those in the latter. We will now proceed to a series of solutions of the Problem of the Three Bodies, or, in the analytical mode of considering the subject, to a series of integrations of the differential equation, vector u = EXAMPLE 1. It is required to find the value of the inverse of the radius when the body, revolving in a circular orbit round an attracting centre or body, is disturbed by the action of a very remote body, which revolves also in a circular orbit, the plane of which is coincident with the plane of the other orbit. The law of the force, whether it be centripetal or disturbing, is supposed to vary according to the inverse square of the distance. The first operation will be to find the value of ПI, and (see pp. 100, 104, &c.) to convert it into a series of cosines of multiples of the arc v. This is a very simple expression for II, obtained on two suppositions: the first, (which does not strictly hold of any case in Nature,) is, that the orbit is circular, and consequently that u is du constant, and = 0: the second, which is nearly true in every dv instance in the planetary system, namely, that the disturbing force |